A259051 Triangle T(n,m) for the number of ways to put n stones into an m X n square grid such that each of the m rows contains at least one stone.

1, 1, 4, 1, 18, 27, 1, 68, 288, 256, 1, 250, 2250, 5000, 3125, 1, 922, 15795, 65880, 97200, 46656, 1, 3430, 105987, 739508, 1932805, 2117682, 823543, 1, 12868, 696864, 7653632, 31539200, 59179008, 51380224, 16777216, 1, 48618, 4540968, 75687696, 461828790, 1320099444, 1919564892, 1377495072, 387420489

Offset: 1

Author

Wolfdieter Lang, Jun 18 2015

Keywords

• nonn
• easy
• tabl

Comments

This is the triangle A258371(n, m)/binomial(n, m).

For the corresponding partition array see A258152.

Examples

			The triangle T(n, m) begins:
n\k 1     2       3        4         5         6           7
1:  1
2:  1     4
3:  1    18      27
4:  1    68     288      256
5:  1   250    2250     5000      3125
6:  1   922   15795    65880     97200      46656
7:  1  3430  105987   739508   1932805    2117682     823543
...
8:  1 12868  696864  7653632  31539200 59179008 51380224
16777216,
9:  1 48618 4540968 75687696 461828790 1320099444 1919564892 1377495072 387420489.
a(4, 2) = 68 from the sum 32 + 36 of the n=4 row of A258152 which belong to the partitions of 4 with m=2 parts, namely (1, 3) and (2, 2).
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)

Crossrefs

Cf. A258152, A258371.

Programs

Mathematica

T[n_, k_]:= Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n, e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n, k], {n, 9}, {k, n}] (* Giovanni Resta, Jun 18 2015 *)

Formula

T(n, m) = sum over the A258152(n, k) entries corresponding to partitions of n with m parts; n >= 1, m = 1,2, ..., n.

T(n, m) = A258371(n, m)/binomial(n, m).